巴拿赫空间结构理论及其在最优化中的应用

Abstract：This research program belongs to the field of the theory of Banach.spaces in functional analysis and its applications in the theory of optimization. The geometry of Banach spaces is an important branch of functional analysis, and vector.optimization is an area where functional analysis, specially, the geometry of Banach spaces plays an important role. In this research program, we mainly study the geometric structure and topological structure of general Banach spaces and some special Banach spaces such as Orlicz spaces, Orlicz-Soblev spaces, and K.ther spaces Using created special technique and methods; we gave the charaterizations of theweak normal structure of Orlicz spaces, isometric copies of 1 l and l∞ , extreme points, strictly convex and uniformly convex properties of Orlicz-Soblev spaces, and H-points of K.ther-Bochner spaces. With the help of the dual mapping of Banach spaces, we generalizes the famous Riesz orthogonal decomposition theorem in Hilbert spaces to general Banach spaces, which make us to give a criterion of orthogonal complementable closed subspaces introduced by James. Consequently, this provides a new criterion of Banach spaces to be isomorphic with Hilbert space. Using the renormed theorem of Banach spaces, we proved that a bounded linear operator in Banach spaces to be a compact linear operator if and only if it can be approximated uniformly by a bounded homogeneous operator. This result displays the essence of counter example given by Enflo. Using the geometric properties, we study systemly Moore-Penrose metric generalized inverse and Tseng metric generalized inverse of linear operator, and single-valued homogeneous selections of set-valued metric generalized inverse, and criteria of its existence, continuity and linear. The result was used to solve non-well.posed problems of partial differential equations and two point boundary problems of ordinary differential equations..Using the topological structure and ordered structure of Banach spaces, we study vector optimization problems and generalized vector equilibrium problems. We discussed the relationships of various proper efficiencies of vector timization, gave out the existence result of generalized vector variational inequalities and vector equilibrium, and proved the arcwise connectedness of efficient point set of a compact.convex set under some conditions. With the help of geometry of Banach spaces, we.provide some characterizations of some remarkable classes of cones, our result give apositive answer to the question proposed by Gong[4].

研究Banach空间的几何与拓扑性质，与不动点相关的一些其他空间性质及子空间同构问题。利用空间的几何和拓扑性质，如凸性、H性质等研究向量最优化问题，有效解集拓扑性质。致跙anach空间上非线性最优化控制问题。进一步揭示出空间的本质属性为解决向量最优化妥钣趴刂浦械哪承┪侍馓岢龈行Х椒ā